Moment equations in the kinetic theory of gases and wave velocities

We consider the evolution system for N-moments of the Boltzmann equation and we require the compatibility with an entropy law. This implies that the distribution function depends only on a single scalar variable which is a polynomial in . It is then possible to construct the generators such that the system assumes a symmetric hyperbolic form in the main field. For an arbitrary we prove that the systems obtained maximise the entropy density. If we require that the entropy coincides with the usual one of non-degenerate gases, we obtain an exponential function for , which was already found by Dreyer. From these results the behaviour of the characteristic wave velocities for an increasing number of moments is studied and we show that in the classical theory the maximum velocity increases and tends to infinity, while in the relativistic case the wave and shock velocities are bounded by the speed of light. Received June 5, 1997     

Maximum velocity for wave propagation in a relativistic rarefied gas

Considering the hyperbolic symmetric system of moments associated with the relativistic Boltzmann-Chernikov equation and closed through procedures of Extended Thermodynamics, we determine numerically the maximum characteristic wave velocity for degenerate and non-degenerate gases. As predicted by recent results, this velocity increases monotonically with respect to the number of moments and tends asymptotically to the speed of light. We show that many moments are often required to approach with a good approximation the speed of light. The dependence of the maximum characteristic wave velocity on the properties of the materials is also investigated and it turns out that it depends significantly on the chemical potential only in the case of Fermions gases. Received April 24, 1999     

Kinetic solutions of the Boltzmann-Peierls equation and its moment systems

The evolution of heat in crystalline solids is described at low temperatures by the Boltzmann-Peierls equation, which is a kinetic equation for the phase density of phonons.In this study, we solve initial value problems for the Boltzmann-Peierls equation in relation to the following issues: In thermodynamics, a given kinetic equation is usually replaced by a truncated moment system, which in turn is supplemented by a closure principle so that a system of PDEs results for some moments as thermodynamic variables. A very popular closure principle is the maximum entropy principle, which yields a symmetric hyperbolic system. In recent times, this strategy has led to serious studies on two problems that might arise: 1. Do solutions of the maximum entropy principle exist? 2. Is the physics that is embodied by the kinetic equation more or less equivalently displayed by the truncated moment system? It was Junk who proved for the BOLTZMANN equation of gases that maximum entropy solutions do not exist. The same failure appears for the Fokker-Planck equation, which was proved by means of explicit solutions by Dreyer, Junk, and Kunik.This study has two main objectives:1. We give a positive existence result for the maximum entropy principle if the underlying kinetic equation is the Boltzmann-Peierls equation. In other words we show that the maximum entropy principle can be used here to establish a closed hyperbolic moment system of PDEs. However, the intent of the paper is by no means a general justification of the maximum entropy principle.2. We develop an approximative method that allows the solutions of the kinetic equations to be compared with the solutions of the hyperbolic moment systems. To this end we introduce kinetic schemes that consists of free flight periods and two classes of update rules. The first class of rules is the same for the kinetic equation as well as for the maximum entropy system, while the second class of update rules contains additional rules for the maximum entropy system. It is shown that if a sufficient number of moments are taken into account, the two solutions converge to each other. However, in terms of numerical effort, the presented solver for the kinetic equation clearly outperforms the one for the maximum entropy principle.Received: 15 August 2003, Accepted: 8 November 2003, Published online: 11 February 2004PACS: 02.30.Jr, 02.60.Cb, 05.30.Jp, 44.10. + i, 63.20.-e, 66.70. + f, 65.40.Gr Correspondence to: M. Herrmann     

Initial Layers and Uniqueness of¶Weak Entropy Solutions to¶Hyperbolic Conservation Laws

We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weak-star in L ^∞ as t→0₊ and satisfy the entropy inequality in the sense of distributions for t>0. We prove that, if the flux function has weakly genuine nonlinearity, then the entropy solutions are always unique and the initial layers do not appear. We also discuss applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation. Accepted: October 26, 1999     

Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws

We consider the problem of self-similar zero-viscosity limits for systems ofN conservation laws. First, we give general conditions so that the resulting boundary-value problem admits solutions. The obtained existence theory covers a large class of systems, in particular the class of symmetric hyperbolic systems. Second, we show that if the system is strictly hyperbolic and the Riemann data are sufficiently close, then the resulting family of solutions is of uniformly bounded variation and oscillation. Third, we construct solutions of the Riemann problem via self-similar zero-viscosity limits and study the structure of the emerging solution and the relation of self-similar zero-viscosity limits and shock profiles. The emerging solution consists ofN wave fans separated by constant states. Each wave fan is associated with one of the characteristic fields and consists of a rarefaction, a shock, or an alternating sequence of shocks and rarefactions so that each shock adjacent to a rarefaction on one side is a contact discontinuity on that side. At shocks, the solutions of the self-similar zero-viscosity problem have the internal structure of a traveling wave.     

On the shock structure problem for hyperbolic system of balance laws and convex entropy

In this paper we give a brief survey of the problem of shock structure solutions in fluid dynamics. For a generic system of balance laws compatible with an entropy principle and a convex entropy we prove that solutions cannot exist when the shock velocity exceeds the maximum characteristic velocity in the equilibrium state in front of the shock. This is in agreement with a conjecture of Extended Thermodynamics. Received February 15, 1998     

Maximum wave velocity in the moments system of a relativistic gas

We consider the system of moments associated with the relativistic Boltzmann-Chernikov equation. Using the particular symmetric form obtained by the closure procedure of Extended Thermodynamics we deduce a lower bound for the maximum velocity of wave propagation in terms of the number of moments for a non-degenerate gas. When the number of moments increases this velocity tends to the speed of light. We also give the lower bound estimate in the limit cases of ultrarelativistic fluids and in the non relativistic approximation. Received September 28, 1998     

Entropy and Global Existence for Hyperbolic Balance Laws

This paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables. We propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state. In addition, we show that a system of balance laws satisfies a Kawashima condition if and only if its first-order approximation, namely the hyperbolic-parabolic system derived through the Chapman-Enskog expansion, satisfies the corresponding Kawashima condition. The result is then applied to Bouchuts discrete velocity BGK models approximating hyperbolic systems of conservation laws.     

Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy

We consider the Cauchy problem for a general one-dimensional n×n hyperbolic symmetrizable system of balance laws. It is well known that, in many physical examples, for instance for the isentropic Euler system with damping, the dissipation due to the source term may prevent the shock formation, at least for smooth and small initial data. Our main goal is to find a set of general and realistic sufficient conditions to guarantee the global existence of smooth solutions, and possibly to investigate their asymptotic behavior. For systems which are entropy dissipative, a quite natural generalization of the Kawashima condition for hyperbolic-parabolic systems can be given. In this paper, we first propose a general framework for this kind of problem, by using the so-called entropy variables. Then we go on to prove some general statements about the global existence of smooth solutions, under different sets of conditions. In particular, the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension. Our main tools will be some refined energy estimates and the use of a suitable version of the Kawashima condition.     

Non-linear extended thermodynamics of real gases with 6 fields

We establish extended thermodynamics (ET) of real gases with 6 independent fields, i.e., the mass density, the velocity, the temperature and the dynamic pressure, without adopting the near-equilibrium approximation. We prove its compatibility with the universal principles (the entropy principle, the Galilean invariance and the stability), and obtain the symmetric hyperbolic system with respect to the main field. In near-equilibrium we recover the previous results. The correspondence between the ET 6-field theory and Meixner׳s theory of relaxation processes is discovered. The internal variable and the non-equilibrium temperature in Meixner׳s theory are expressed in terms of the quantities of the ET 6-field theory, in particular, the dynamic pressure. As an example, we present the cases of a rarefied polyatomic gas and study the monatomic-gas limit where the system converges to the Euler system of a perfect fluid.     

A stochastic system of particles modelling the Euler equations

We consider a system of N spheres interacting through elastic collisions at a stochastic distance. In the limit N , for a suitable rescaling of the interaction parameters, we prove that the one-particle distribution function converges to a local Maxwellian, whose gross density, velocity, and temperature satisfy the Euler equation.     

Equilibrium states of an elastic conducting rod in a magnetic field

Our principal concern is an analysis of the equilibrium states of a nonlinearly elastic conducting rod in a magnetic field. We assume hyperelasticity so the equilibria formally appear as critical points of a potential energy functional on the strains. Fairly standard methods give existence of a minimum (not necessarily unique) with e.g., L₂-regularity. The assumptions imposed on the functional preclude the use of the usual techniques for justification of the formal necessary conditions for optimality. A new general technique is developed to justify these conditions; it then follows that minimizers satisfy the equilibrium conditions in the classical sense. (A feature of this technique is that the variations considered are homotopies so one can consider minimization within a homotopy class.) In the symmetric case, which admits trivial (straight and untwisted) solutions, we show that nontrivial solutions also exist if the field is strong enough.     

Global BV Entropy Solutions and Uniqueness for Hyperbolic Systems of Balance Laws

We consider the Cauchy problem for n×n strictly hyperbolic systems of nonresonant balance laws each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that and are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation as limits of special wave-front tracking approximations for which the source term is localized by means of Dirac masses. Moreover, we give a characterization of the resulting semigroup trajectories in terms of integral estimates.     

Energy Flow in Formally Gradient Partial Differential Equations on Unbounded Domains

As an example of an extended, formally gradient dynamical system, we consider the damped hyperbolic equation u _tt+u _t=u+F(x, u) in R ^N , where F is a locally Lipschitz nonlinearity. Using local energy estimates, we study the semiflow defined by this equation in the uniformly local energy space H¹ _ul(R ^N )×L² _ul(R ^N ). If N2, we show in particular that there exist no periodic orbits, except for equilibria, and we give a lower bound on the time needed for a bounded trajectory to return in a small neighborhood of the initial point. We also prove that any nonequilibrium point has a neighborhood which is never visited on average by the trajectories of the system, and we conclude that any bounded trajectory converges on average to the set of equilibria. Some counter-examples are constructed, which show that these results cannot be extended to higher space dimensions.     

Entropy principle for the moment systems of degree $\alpha$ associated to the Boltzmann equation. Critical derivatives and non controllable boundary data

m moments and of degree . For such theories the entropy principle is still valid only, if the non equilibrium field variables and their derivatives are sufficiently small with respect to the required approximation order. In this paper we prove through simple examples of stationary problems that the entropy principle fails in general, if all the non-equilibrium variables are of the same order of magnitude. This is due to the fact that there exist some derivatives of non equilibrium variables (critical derivatives) that are not small along all the solutions. This property can be used to fix the non controllable boundary data in such a manner that the critical derivatives are kept small for the solution that we may choose. Thus, for the stationary unidimensional case we propose the requirement that the critical derivatives vanish on the boundary, eventually with some successive derivatives. This is a sufficient condition for the validity of the entropy principle at least in a neighborhood of the boundary and makes it possible to assign the non controllable data in a simple manner when the number of moments is greater than 13. We have tested this procedure in several cases of theories, showing that the criterion implies continuity with respect to the change of the moment number and to the truncation order. In particular for the planar unidimensional heat conduction problem we have obtained a behavior for the temperature that is always the same as the one predicted by the classical Fourier law. This result is in evident contrast with the minimax principle expectation. However we have qualitative differences between the temperature behavior described by Extended Thermodynamics and the one by Fourier-Navier-Stokes theory for heat conduction in radial symmetry. Received June 25, 2001 / Published online February 28, 2002     

Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$

This paper is devoted to a class of nonautonomous parabolic equations of the form u _t = Δu + f(t, u) on \mathbbR^N{\mathbb{R}^N} . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.     

Large-Time Behavior of Solutions to Hyperbolic-Elliptic Coupled Systems

We are concerned with the asymptotic behavior of a solution to the initial value problem for a system of hyperbolic conservation laws coupled with elliptic equations. This kind of problem was first considered in our previous paper. In the present paper, we generalize the previous results to a broad class of hyperbolic-elliptic coupled systems. Assuming the existence of the entropy function and the stability condition, we prove the global existence and the asymptotic decay of the solution for small initial data in a suitable Sobolev space. Then, it is shown that the solution is well approximated, for large time, by a solution to the corresponding hyperbolic-parabolic coupled system. The first result is proved by deriving a priori estimates through the standard energy method. The spectral analysis with the aid of the a priori estimate gives the second result.     

Properties of shock adiabats in the neighborhood of jouguet points

Two well-known properties of shock adiabats in a gas [1] are proved for shock adiabats corresponding to discontinuous solutions of hyperbolic systems of equations expressing conservation laws. If the state on one side of a discontinuity is fixed, then at the point of extremum of the discontinuity velocity on the shock adiabat the velocity of the discontinuity is equal to one of the velocities of the characteristics on the other side of the discontinuity and vice versa. If for the systems there is defined an entropy flux or mass density of entropy, then at the points of extremum of the velocity there is an extremum of the entropy production at the discontinuity and the entropy mass density. If the system is a symmetric hyperbolic system [2, 3], then the extrema of the entropy production at the discontinuity correspond to extrema of the velocity. These properties may be helpful in the study of discontinuities in complex media, since the sections of a shock adiabat whose points can correspond to actually existing discontinuities are frequently bounded by points corresponding to discontinuities whose velocity is equal to the velocity of a characteristic on one of the sides of the discontinuity (see, for example, [1, 4, 5]).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 184–186, March–April, 1979.     

Second law analysis of the 2D laminar flow of two-immiscible,incompressible viscous fluids in a channel

The laminar and fully developed flows of two immiscible fluids confined in a thin slit of constant wall heat fluxes are analyzed in terms of entropy generations due to irreversibility of forced convection heat transfer. The governing equations are analytically derived using expressions for velocity distributions. The derived equation for the dimensionless entropy generation number is used to interpret the relative importance of frictions to conduction by varying irreversibility distribution ratio ϕ. It is found that the minimum entropy generation takes place at the dimensionless half transverse distance (ξ) of 0.3 for values of ϕ higher than zero. The entropy generation near the plate increases more rapidly in fluid I than in fluid II as viscous dissipation effects becomes more important. The velocity profiles are found to be in agreement with the distributions of the dimensionless entropy generation number (N _S ) for two-immiscible incompressible flows in the slit.     

Binary Decompositions for Planar N-Body Problems and Symmetric Periodic Solutions

A binary decomposition for a system of N masses is a way of treating the system as binaries with the total action exactly the same as that of the original system. By considering binary decompositions, we are able to provide effective lower-bound estimates for the action of collision paths in several spaces of symmetric loops. As applications, we use our estimates to prove the existence of some new classes of symmetric periodic solutions for the N-body problem.